Properties of fuzzy set: All at one place

Properties of fuzzy set help us to simplify many mathematical fuzzy set operations. Sets are collections of unordered, district elements. We can perform various fuzzy set operations on the fuzzy set. It is recommended to the reader to first navigate through the fuzzy set operations for a better understanding of the properties of the fuzzy set.

Most of the properties of crisp sets are held for fuzzy sets also.

Note: To differentiate fuzzy sets from the classical set, we will be putting a bar under or over the set notation

Suggested Reading: You may refer the article on Fuzzy Terminology to understand the technical terms associated with Fuzzy Logic

Properties of Fuzzy Sets:

Involution

Involution states that the complement of complement is set itself.

( A‘ )’ = A

Commutativity

Operations are called commutative if the order of operands does not alter the result. Fuzzy sets are commutative under union and intersection operations.

A ∪ B = B ∪ A

A ∩ B = B ∩ A

Associativity

Associativity allows change in the order of operations performed on an operand, how ever relative order of the operand can not be changed. All sets in the equation must appear in identical order only. Fuzzy sets are associative under union and intersection operations.

A ∪ ( B ∪ C ) = ( A ∪ B ) ∪ C

A ∩ ( B ∩ C ) = ( A ∩ B ) ∩ C

Distributivity

A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C )

A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C )

Absorption

Absorption produces identical sets after stated union and intersection operations.

A ∪ ( A ∩ B) = A

A ∩ ( A ∪ B ) = A

Idempotency / Tautology

Idempotency does not alter the element or the membership value of elements in the set

A ∪ A = A

A ∩ A = A

Identity

A ∪ ϕ = A

A ∩ ϕ = ϕ

A ∪ X = X

A ∩ X = A

Transitivity

If A ⊆ B and B ⊆ C then A ⊆ C

De Morgan’s Law

De Morgan’s Laws can be stated as,

• The complement of a union is the intersection of the complement of individual sets
• The complement of an intersection is the union of the complement of individual sets

( A ∪ B )’ = A‘ ∩ B‘

( A ∩ B )’ = A’ ∪ B‘

Watch on YouTube: Properties of fuzzy set

Let a = μ_A(x) and b = μ_B(x), we can define the properties of the following operations as,

Fuzzy Complement

C:[0, 1]→[0, 1], which satisfies the following axioms

• Axiom 1: C(0) = 1, C(1) = 0 (boundary condition)
• Axiom 2: If a < b, then c(a) ≥ c(b)
• Axiom 3: C is continuous
• Axiom 4: C(C(a)) = a

Axiom 1 and Axiom 2 form Axiomatic Skeleton for a fuzzy complement

Fuzzy Union

U:[0, 1]×[0, 1]→[0, 1]

• Axiom 1: U(0, 0) = 0, U(1, 0) = 1, U(0, 1) = 1, U(1, 1) = 1   (Boundary Condition)
• Axiom 2: If a < a′ and b < b′ then U(a, b) ≤ U(a′, b′)  (monotonic)
• Axiom 3: Commutative: U(a, b) = U(b, a)
• Axiom 4: Associative: U(U(a, b), c) = U(a, U(b, c))
• Axiom 5: U is continuous
• Axiom 6: U(a, a) = a  (Idempotency)

Axioms 1 to 4 form Axiomatic Skeleton for fuzzy union

Fuzzy Intersection

I:[0, 1]×[0, 1]→[0, 1]

• Axiom 1: I(0, 0) = 0, I(1, 0) = 0, I(0, 1) = 0, I(1, 1) = 1   (Boundary Condition)
• Axiom 2: If a<a′ and b<b′ then I(a, b) ≤ I(a′, b′ )    (monotonic)
• Axiom 3: Commutative: I(a, b) = I(b, a)
• Axiom 4: Associative: I(I(a, b), c) = I(a, I(b, c))
• Axiom 5: I is continuous
• Axiom 6: I(a, a) = a  (Idempotency)

Axioms 1 to 4 form Axiomatic Skeleton for fuzzy intersection